To solve the equation \(2 - |x + 3| = \frac{1}{4}\), we can start by isolating the absolute value expression. Here are the steps to solve it:

1. **Isolate the absolute value:**

\[
|x + 3| = 2 - \frac{1}{4}
\]

Simplifying the right side, we get:

\[
|x + 3| = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}
\]

2. **Set up the cases for the absolute value:**

The absolute value equation \( |x + 3| = \frac{7}{4} \) gives us two scenarios:

**Case 1:**
\[
x + 3 = \frac{7}{4}
\]

**Case 2:**
\[
x + 3 = -\frac{7}{4}
\]

3. **Solve each case:**

**For Case 1:**
\[
x + 3 = \frac{7}{4}
\]
\[
x = \frac{7}{4} - 3 = \frac{7}{4} - \frac{12}{4} = -\frac{5}{4}
\]

**For Case 2:**
\[
x + 3 = -\frac{7}{4}
\]
\[
x = -\frac{7}{4} - 3 = -\frac{7}{4} - \frac{12}{4} = -\frac{19}{4}
\]

4. **Final solutions:**

Thus, the solutions to the equation \(2 - |x + 3| = \frac{1}{4}\) are:

\[
x = -\frac{5}{4}, \quad x = -\frac{19}{4}
\]